Suggested Citation:"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990. The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/1604.

Suggested Citation:"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990. The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/1604.

Suggested Citation:"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990. The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/1604.

Suggested Citation:"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990. The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/1604.

Suggested Citation:"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990. The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/1604.

Suggested Citation:"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990. The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/1604.

Suggested Citation:"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990. The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/1604.

Suggested Citation:"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990. The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/1604.

Suggested Citation:"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990. The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/1604.

Suggested Citation:"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990. The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/1604.

Suggested Citation:"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990. The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/1604.

Suggested Citation:"Simulation of Hydrodynamic Loading and Structural Response of a Marine Riser." National Research Council. 1990. The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/1604.

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Simulation of the Hydrodynamic Loading and Structural Response
of a Marine Riser
A. Dercksen and F. van Walree
Maritime Research Institute Netherlands
Wageningen, The Netherlands
Abstract
The flow around circular cylinders at moderate
to high Reynolds numbers is characterized by vortex
shedding. Hydroelastic oscillations are caused by the
oscillatory in-line and transverse forces, induced by
this periodic shedding of vortices. The hydrodynamic
loading and structural response of a marine riser is
simulated using the results of several flow simulations
with the vortex blob method. A three-dimensional
vortex blob method is discussed. Operator splitting
is used to obtain a formulation in which the viscous
effects are incorporated using a stochastic process,
while the inviscid flow is described by the evolution
of a flow map. The numerical algorithm consists of an
accurate and efficient spectral model combined with
a variational formulation. A comparison with exper-
imental results and semi-empirical models is used for
validation. A description of the computational as-
pects related to the coupling of the results obtained
from the vortex blob method to an existing struc-
tural analysis code is given. Finally some computed
results of the dynamic response of a flexible riser, a
pipe and a cable subjected to environmental forces
due to current will be compared with experimental
and full scale results.
1. Introduction
Hydroelastic oscillations are caused by the peri-
odic shedding of vortices from the boundary layer
of a flexible cylindrical blunt body, inducing trans-
verse and in-line forces. The vortex shedding fre-
quency may lock on to the natural frequency of the
system, provided the internal structural damping is
sufficiently low. The structure extracts energy from
the flow resulting in a motion amplitude increase to
values of once to twice the characteristic length. An
important issue is the increased drag force and risk of
fatigue due to the high frequency oscillations of the
structure.
Several semi-empirical formulae exist for the pre-
diction of drag and lift forces. These models all have
in common that details of the flow field are not taken
into account, i.e. they are chiefly based on mechani-
cal and electrical analogous. More advanced models
contain tuning parameters to fit the model to exper-
imental observations. Again, these parameters have
no physical interpretation other than their analogous
in other fields. On the contrary, very advanced nu-
merical schemes exist for solving the instationary vis-
cous Navier Stokes equations. These methods cer-
tainly take into account the flow field, but even with
present day computing power such complex simula-
tions will impose an economical limit on their use for
practical problems.
The authors stress that these models are invalu-
able for research purposes such as understanding more
about the details of such complex flows. Attempts
have been made to couple the direct flow simula-
tion to a structural code by e.g. Hansen et al. ~l].
The limitations on computer power require many sim-
plifications and heuristic arguments in the model.
The predictive abilities of this approach are there-
fore rather doubtful. For practical situations such as
hydroelastic oscillations, simplified models must be
developed, drawing on the results and insights ob-
tained from large scale, systematic computations and
experiments. As a first step in this direction, a wake-
oscillator model (e.g. Griffin 2], j3],[44) is tuned to
the results obtained by performing a number of sim-
ulations using a vortex blob method for solving the
Na~rier-Stokes equations. The calculations were con-
ducted for a two-dimensional flow around a spring
mounted circular cylinder. The tuned model is subse-
quently used in a structural analysis code to simulate
the hydrodynamic loading and structural response of
a marine riser, a pipe and a cable in current.
617

2. Theory Vortex Blob Method rR(r', t, to)
The equations governing the evolution of a vortic-
ity field A, defined as the curl of the velocity field A, in
a viscous incompressible fluid in a Eulerian reference
frame are commonly known as the viscous vorticity
transport equations. They can be summarized as:
D~-=~-Vu+Z'V2w (1)
with the incompressibility constraint V · u = 0 for
the velocity field. In addition the velocity field must
satisfy the no-slip boundary condition at a smooth
body surface S and approach a uniform velocity field
U far ahead of the body. It is therefore advanta-
geous to define the total velocity field u as the sum
of two components, viz. u = u + use with u the dis-
turbance velocity field which approaches a zero value
far away from the body, and use the velocity field
of the irrotational flow around the body. The set of
equations is completed with an initial vorticity field
we which is assumed to be tangential to the body
surface. The left-hand side of the vorticity trans-
port equation is the material derivative of the vor-
ticity field, whereas the right-hand side is responsible
for vortex stretching and diffusion of vorticity. Note
that in two-dimensional flows, the stretching term is
absent.
The vortex blob method is based on operator split-
ting, where all relevant physical processes are mod-
elled subsequently in time. The solution of the vis-
cous vorticity transport equations is approximated by
successively solving the inviscid vorticity transport
equations for a small time step, followed by the dif-
fusion of vorticity and the creation of vorticity at the
body surface, in order to maintain the no-slip con-
dition. Several authors have investigated this frac-
tional step algorithm as was proposed by Chorin, e.g.
Chorin t5], Beale [6],~7],[8~. Recently, Van der Vegt t94
has given a convergence proof of a product algorithm
for the approximation of the solution of the viscous
vorticity transport equations.
In order to state the algorithm, it is necessary to
introduce some definitions. The inviscid flow map R
is defined as an invertible continuously differentiable
mapping from the initial fluid volume DO to the fluid
volume Q at time t:
R(r',t,tO`): TO ~ Q At ~ [to,t~] (2)
with r' the position of a fluid particle at initial time
to. The position r of a fluid particle at time t then is
given by the relation:
618
(3)
If r' is kept fixed, while t varies, equation (3) reps
resents the path of a fluid particle initially at A'. The
flow map R must have a Jacobian determinant unity
due to the incompressibility constraint, t104. Anal-
ogous to the velocity field u it is advantageous to
separate the total flow map R in a part R - , related
to the velocity field of the potential flow around the
body and a disturbance flow map R by means of the
relations:
R(r',t,tO) = R°° 0 R(r',t,tO) (4)
with:
Rot (r', t, to) = r' + use (tto) (5)
where up is the velocity field for the irrotational flow,
i.e. the potential flow.
The vector stream function A can be defined by
relating the curl of the vector stream function to the
total velocity field n. The use of the vector stream
function has the advantage that this representation
immediately satisfies the incompressibility constraint
for the flow field. The relation between the fluid par-
ticle velocity R and the Eulerian field velocity u now
can be expressed as:
R (r', t, to) = Vr x A (r, t)
= VR X A (R (r', t, to), t) (6)
The relation between the vorticity field in both co-
ordinate systems can be constructed likewise, [10],
yielding:
~ (r, t) = me (a') Vr`R (r', t, to) (7)
The introduction of the flow map has transformed
the problem of determining the vorticity field by solv-
ing the inviscid vorticity transport equation into the
calculation of the flow map R and the vector stream
function A. The system of equations (6) and (7)
must be completed by deriving equations for the vec-
tor stream function. This can most conveniently be
done by first separating the vector stream function A
into several components:
A = A + Am + V?,b
(8)
where the vector stream function Am has a curl equal
to up. The function fib which must be twice differ-
entiable can always be chosen such that the distur-
bance vector stream function A is divergence free. It
is, however, not necessary to calculate the function fib

because in the vortex model only the curl of the total
vector stream function is required, hence the gradi-
ent in equation (8) will give no contribution to the
velocity field.
The relation between the vorticity field 5~0 and the
disturbance vector stream function A now is obtained
by using the definition of the vorticity field and the
divergence freedom of A:
V2A (`r, t) =~ (or, t) (9)
The boundary conditions for the vector stream func-
tion A can be directly obtained from the no-flux con-
dition for the velocity field at the surface S in an
inviscid fluid and equation (8), yielding:
n. V x A =n. V x Am = 0 at S (10)
since An is the stream function for the potential flow
around the cylinder. At great distance from the body
the disturbance vector stream function A must sat-
. ,
ashy:
A ID
(11)
The whole system of equations for the disturbance
flow map, vector stream function and vorticity field
(6), (7) and (9) was put by Van der Vegt t9] in a
Lagrangian variational formulation for bodies with an
arbitrary geometry by means of the following Action
Principle:
J [Lr' A, to, try = J dtL [R. A, t] (12)
with Lagrangian functional:
L fR, A, t] = ~ d3r(~0 (or'`) Vr`R (r', t, to)
(~-R (T., t, to) x R (r', t, to)A (R. t))
+ - ~ d3T TV X A (r t) ~2 (13)
2 n
together with the initial condition n · 5~0 at S for the
vorticity field and boundary condition n · V x A = Q
at S for the vector stream function yields the set of
equations for the flow map and vector stream function
after variations bR and bA, with t · [A = Q at the
body surface S with tangential vector t. Here the
integration in the Lagrangian functional is conducted
over both the initial fluid volume SO as well as the
fluid volume Q at time t.
The Eulerian-Lagrangian formulation requires at
first sight the determination of both a disturbance
flow map and a vector stream function after which
the vorticity field can be determined. The distur-
bance flow map can, however, be obtained from the
Taylor series expansion of the flow map. This is ac-
complished by deriving from the Lagrangian formula-
tion a Hamiltonian formulation using a Dirac bracket.
An extensive discussion of this reduction process can
be found in t94. The result, which can be verified by
using (7), reads:
R (`r', t +^t, t) = R (T', t, t) +
At (VR X A (R. t) ) +
1t,t2 (VR X A (R,t)) VR (VR x A (R't))
+(9 (^t3)
(14)
The determination of the flow map R is the first
step in the product formula. The next step is the
modelling of the effects of viscosity, i.e. diffusion and
creation of vorticity. The model is based on the ran-
dom walls interpretation of the incompressible Navier-
Stokes equations given by Peskin t114. The effect of
viscosity is to generate a random disturbance on the
particle paths generated by the flow map R. The
stochastic flow map is defined as the sum of the in-
viscid flow map and a disturbance flow map which is
a random vector from a sphere of radius (~12~T)2, ~
being a small time step. The viscous vorticity evolu-
tion operator is represented by the following sequence
of maps:
~ ~ ~ ° ET ° DOT (15)
where ~ is an operator which reflects particles, which
diffuse into the body, across the boundary. The invis-
cid evolution operator En is defined by equation (7~.
The diffusion operator DO causes a random transla-
tion of each point of the vorticity field.
The main result of Van der Vegt t9; states that,
provided the inviscid flow map exists and is unique,
the expectation of the viscous evolution operator con-
verges to the evolution operator of the viscous vor-
ticity transport equations. Analogously, the expecta-
tion of the particle paths, converge to the real particle
paths. More details and a proof of convergence can
be found in Van der Vegt t94.
In the next section, the numerical implementation
of the vortex blob model is discussed.
3. Numerical Implementation
_ . .. . .. . _ . .. . . . .. ..
The numerical implementation of the vortex model
described in the previous section requires the dis-
cretization of the vorticity field and an algorithm to
619

evaluate the flow map. For two-dimensional flows
several schemes have been suggested. In the point
vortex method the vorticity field is represented by a
set of point vortices as follows:
N
w(r,t) = ~ri6(r_i(t)) (16)
i=1
where N is the number of vortices, r, represents the
trajectory of the vortex with index i.
The flow map is then calculated by means of the
Biot-Savart law of interaction, resulting in the fol-
lowing set of ordinary differential equations for the
particle trajectories:
 ~ ~ it ~) z · +B (but) (17)
Here _B is an additional velocity due to an onset flow
and the disturbance velocity caused by the body. The
advantage of this procedure is that it is gridless, so
that numerical problems related to grid generation
and stability criteria for high Reynolds numbers are
avoided. However, the algorithm has an operation
count proportional to N2, and suffers from singular
behaviour in the induced velocities when two vortices
approach each other. The singularities lead to chaotic
motions after some time steps. A variation to the
point vortex method is the vortex in cell method in
which the flow map is calculated on a fixed grid, re-
ducing the singular behaviour and gaining numerical
efficiency at the cost of accuracy.
The discretization used in this paper is the vortex
blob approximation:
N
~ (r, t) = ~ rid (lori (t)l) (18)
i=1
where by is a function of compact support e.g. a Gaus-
sian distribution. This method is more suitable for
flows with a smooth vorticity field, and eliminates the
singular behaviour in the induced velocities.
The vorticity field obtained by this discretization,
however, does not satisfy the inviscid transport equa-
tions. Convergence criteria for the two-dimensional
vortex blob methods are given by Hald et al. [12i,
t134. In order to calculate the trajectories of the
blobs, the stream function is needed. As was men-
tioned earlier, the use of Biot-Savart's law is numer-
ically inefficient. Improvements are made by solving
the Poisson equation for the stream function on a grid
fixed in space. This results in an operation count pro-
portional to the number of vortex blobs N. plus some
overhead for a fast Poisson solver.
The procedure discussed in this paper is based
on the results from Section 2. For two-dimensional
flows, the stream function has only one non-zero com-
ponent. The stream function is separated into three
parts:
A=Ah+AP+A~ (19)
where AP is the solution of the Poisson equation in an
unbounded domain, Ah is the solution of the Laplace
equation with boundary conditions at the body sur-
face S. and Am has a curl equal to the uniform onset
velocity U°°:
V2AP = _~d
v2Ah = 0
n.(VxA e*) =
(20)
(21)
 n.~(V x APez)a,,, + U - ) (22)
The velocity of the surface 5 is given by u'`,. Both
stream functions AP and Ah must have a curl equal to
zero at infinity. Substituting AP and ~ in the action
principle, results in:
J [rj,A ] = 2/ dt/dr'~VAP(~')~2
[i / dt ~j / dr ~ (I fir'~ (t)~) AP (r')
2 ( `' j ~ )}
in which rj = (Xj,Yj) is the trajectory of a vortex
blob with index j in an unbounded fluid. The stream
function AP is expanded in Fourier harmonics with
period Lz and Ly in x and y-direction:
AP (`r'`) = As, Ak exp (ill r') (24)
k
Substituting this relation in (23) and taking vari-
ations with respect to A`, Xj and Yj leads to the
following set of equations:
LzLy Skid = p (I) ~ rj exp ~ik · r) (25)
dXj ~ P (akin ~ At exp (ill
· r) (26)
clip = '9X P (skip ~ Ak exp (`ik r) (27)
where the filter function P is defined as:
P(lkl)= /
620
I dr'~ (fir' - ~ (t)~)exp (id 7.') (28)

For each vortex, the number of trigonometric func-
tions to be evaluated equals the number of vortices
N. This can be avoided by employing cubic spline ap-
proximations to the exponentials, which has the extra
advantage of being able to use Fast Fourier Trans-
forms. Several of the spline approximations were
tested, and are discussed in Van der Vegt et al. [14~.
After calculating the stream function AP, the bound-
ary conditions for Ah are immediately known, and Ah
can be determined. The paths of the vortices depend,
however, only on the curl of stream function so that
it is more efficient to formulate the problem in terms
of Oh = V x Ahez. Bearing in mind the product for-
mula for the solution of the Navier-Stokes equations,
and the fact that the only way to introduce vorticity
in a flow is through the no-slip condition at a solid
boundary, the following integral formulation was de-
rived:
P /s of (tip · K2 (P. q)) dSq =
/s (I (u,,,q UV X Aqez))
(tp Kit (`p,q`)) dSq (~29)
in which lye is the strength of a vortex layer over the
body surface S. t is the tangential vector on the sur-
face, and ]~i,~2 are given by:
Kit (p, q) = Vpln |T<P~ | (~30~)
K2 (`P, q`) = ez x Kit (`p, q) (~31)
More details on the derivation of the boundary in-
tegral representation can be found in e.g. Hunt t15~.
The integral equation for fly' does not have a unique
solution (Hunt, t154~. When the body surface is dis-
cretized by panels with a constant source strength
and vortex strength, this leads to a matrix which is
nearly singular. Since vorticity can only be intro-
duced through the no-slip condition, the integral of
w. over the surface must be equal to zero:
/~y~dS = 0
(32)
Adding this condition to the boundary integral
equation poses a problem with a unique solution. The
inviscid vorticity transport can now be summarized
ate follows:
R = uP + ?,,h + U (33)
which is solved using a second order Runge-Kutta
method. The resulting tangential velocity jump (vor-
ticity layer) at the surface, is represented by a set of
blobs, which are diffused into the fluid through appli-
cation of the random walk.
The two-dimensional model as presented here is
discussed in more detail in Van der Vegt et al. [9i,
t16i, t144. More information about two-dimensional
methods for solving the viscous transport equations
can be found in e.g. Sarpkaya t17i, Tiemroth t18] and
Leonard [194. For simulation of high Reynolds num-
ber flows, many blobs and small time steps are re-
quired, resulting in a large computational effort. The
extension of the vortex blob model to three dimen-
sions is not straightforward. Due to the large number
of vortices required to obtain a realistic description
the flow field, the tracking of individual vortices is
not feasible. Furthermore the stretching term in the
viscous vorticity evolution equation introduces extra
difficulties. For the three-dimensional flows, a more
advanced computational scheme is required. Van der
Vegt t9] developed a special purpose spectral model
to calculate the flow map. The discretization of the
vorticity field also requires special attention. A new
representation of the vorticity, employing blobs which
are modified to account for the presence of a body, is
presently under development. Results and theoretical
details will be published in the near future.
4. Vortex Induced Cylinder Oscillations
The wake of a bluff body is comprised of an al-
ternating vortex street. The shedding frequency fv
of the vortices is a function of the ambient flow ve-
locity U. cylinder diameter D and Reynolds number.
This relation was first discovered by Strouhal. The
non-dimensional Strouhal number S is defined as:
find
S= U
(34)
and is more or less constant over a wide range of
Reynolds numbers (100 < Re < 10000): S = 0.2, for
smooth cylinders. If the cylinder is flexibly mounted,
there are non-linear interactions between the shed-
ding frequency and the cylinder motion. Provided
the damping is suffiently low, the cylinder can extract
energy from the flow, exciting sustained oscillations
at a frequency close to, or coincident with, its natural
frequency. In water both cross-flow and and in-line
excitation occurs. The in-line oscillations are excited
at much lower velocities than required for cross-flow
excitation.
Two dimensionless numbers which are often used
in analyzing experimental data are the reduced ve-
locity:
I/; = U/fnD
and the stability parameter:
(35)
Ks = 2mi/pD21 (36)
621

where In is the natural frequency of the spring mounted
cylinder, m is the virtual mass, ~ is the logarithmic
decrement of damping, p the fluid density and I the
cylinder length. Experiments have shown that the
cross-flow excitation range extends over 4.5 < Vr <
10 with a maximum amplitude of 1.5 diameters. For
in-line oscillations there are two instability regions
within 1.25 ~ Vr ~ 3.8 with a maximum amplitude
of 0.20 diameters. More details on experimental re-
sults can be found in the review of Sarpkaya t204.
During a test program conducted at MARIN, mea-
surements were performed on the spring mounted cylin-
der. A general arrangement including the co-ordinate
system is presented in Fig. 1.
~ Spring
-
Uni f orm
on set
flow
1
1
-
1 ~ x
Fig. 1. Spring mounted cylinder arrangement
Cross-flow and in-line oscillation experiments were
performed. A comparison with the semi-empirical
wake-oscillator as proposed by Griffin is discussed
in [224. The results appeared very promising, and
are rendered valuable for validating the vortex blob
method. The two-dimensional vortex blob code has
been tested on a number of cases of practical inter-
est. Calculations on a fixed cylinder and on a cylinder
forced to oscillate were presented in t164 and [21~. For
the spring mounted cylinder, the method was supple-
mented with a module for solving the equation of
motion of a spring mounted cylinder. At each time
step, the exciting force originating from the flow field
is used as the forcing term for the equation of mo-
tion of the cylinder, which is then displaced, causing
a modification of the flow field. The region which is
of practical importance is the lock-in region, where
the vortex shedding frequency is near the natural fre-
quency of the mass-spring system. The added mass
and damping are incorporated in the calculation of
the forces from the vorticity field. The equation of
motion is calculated using a second order Runge-
Kutta method in time, as is the case for the update
of the blob positions.
5. Structural Analysis Code
The structural analysis code available at MARIN
is a general purpose time domain simulation program
to compute the three-dimensional behaviour of sub-
merged cylindrical bodies excited by end motions,
waves and current. Although the program was de-
veloped for flexible risers, it may also be applied to
other submerged flexible slender structures, such as
Bowline bundles, hoses, umbilicals and mooring lines.
The mathematical model is based on a discrete el-
ement technique known as the lumped mass method.
This technique involves the lumping of mass, excita-
tion forces and reaction forces at a finite number of
nodes along the structure. All forces are formulated
in terms of element properties, i.e. position and ori-
entation. By formulating the laws of dynamic equi-
librium and the stress-strain relations for each node,
a set of equations of motion results. Additional equa-
tions are derived for the element twist motions due to
torsion. These equations are solved in the time do-
main using finite difference and iterative procedures.
It is assumed that the structural elements have ax-
isymmetrical properties with respect to dimension,
fluid force coefficients and stiffness.
A detailed description of the computer program
can be found in Van den Boom et al. t234. In this
paper the discussion is limited to the incorporation
of vortex induced fluid forces and motions in the
model. At each time step the velocity and acceler-
ation vectors of the structural elements are known.
From these vectors, the cross-flow velocity and ac-
celeration components are determined. Using these
quantities, equation (37) is solved using a second or-
der Runge-Kutta method. The cross-flow hydrody-
namic lift force is then known. The drag force is
derived from equation (40) and the time history of
the structural response. The lift and drag forces are
then evenly distributed along each element. From
the excitation forces and reaction forces, the program
computes the kinematics for the next time step. This
process is repeated for each time step. A circular
cross-section is assumed in this approach, i.e. ele-
ment rotation is assumed to be perpendicular to the
flow. The flow along each element is further assumed
622

to be fully correlated, i.e. two-dimensional. This
assumption is valid as the amplitude of motion in-
creases t204. Although the fluid loading is thus es-
sentially two-dimensional on each element, the struc-
tural response computational scheme allows for three-
dimensional responses.
6. Wake-Oscillator Models
As an attempt to collect all observed phenom-
ena in a compact model, several semi-empirical ap-
proaches exist [20~. One of the models is the wake-
oscillator model, in which a non-linear oscillator for
the lift force is coupled to the equation of motion of
the cylinder. The response parameter SG = 27rS2K5
is the main parameter of importance. The governing
equations of the so-called Griffin Model are:
Cr + DOCK - ECHO - Cr - (Cr/ws) ~
[wsGCrh~sHCr] = ~sFY/D (37)
for the lift coefficient, and:
y + 2t,~ny + w2y = (pU2D/2m) CE (38)
for the equation of motion in the direction normal to
the onset flow. The shedding frequency us follows
from the Strouhal relation and the natural frequency
from an = ^, where k is the spring stiffness.
The dimensionless coefficients F. G and H are to be
determined from experimental results.
The major drawbacks of the model are the facts
that it is based on fluid damping in still water and
that there is a continuous phase angle variation be-
tween the exciting force and the response of the cylin-
der. Additional damping has therefore been added to
the Griffin model. This damping is a function of the
cross-flow displacement and is given for By > 0.25
by:
where:
4m [~/4 + 0 25 (by0 25)] (39)
This is a slightly modified form of the empirical equa-
tion as proposed by Skop et al., see e.g. Sarpkaya [204.
For the drag coefficient of the oscillating cylinder,
the following empirical formulation is used:
CD = CD + J sin(2wt) (40)
CD = CDO (1 + I (~y/D) ) (41)
where h' is the frequency of cross-flow motion, By is
the standard deviation of the cross-flow motion and I
and J are constants to be determined from numerical
results obtained with the vortex blob method. CDO
is the mean drag coefficient for a stationary cylinder,
Red = u~nD2/z' is the Reynolds number based on the
oscillation.
7. Discussion of Vortex Blob Calculations
Calculations using the vortex blob method, were
performed for values of the reduced velocity ranging
from 5.0 to 7.0. The Reynolds number ranged from
0.9 x 105 to 1.2 x 105. Table 1 shows a comparison be-
tween experimental and computational results for the
cross-flow motion of a spring mounted rigid cylinder,
with Vr = 5.25, and D=0.1 m:
Table 1
Cross-flow results: measured and calculated
Quantity
_
CD
acD
~CD
CLma:e
~C,.
ACT,
Oman
my
. my
Measured
2.26
1.10
18.33
6.77
3.83
9.25
1.27
0.79
9.25
Computed
At* - 0.05
3.49
0.59
19.60
4.40
1.70
9.75
0.20
0.13
9.80
l At* = 0.005
4.50
0.52
19.60
8.33
2.50
9.75
0.22
0.10
9.80
The most striking result is the difference between
the maximum amplitudes of the motion. The cal-
culated value seems to be closer to normal values re-
ported in literature, while the measured value belongs
to the largest values reported. As was mentioned be-
fore, the calculations were not performed to get con-
clusive numerical results, but to capture the relevant
phenomena involved. The calculated quantities show
a reasonable resemblance with the measured data. Of
course the comparison above does not take into ac-
count the details of the flow field. The vortex blob
algorithm contains parameters, some of which must
be determined through numerical tests. Interesting
items in the calculations were:
· start-up time, i.e. dimensionless time passed
before the cylinder was allowed to oscillate,
· time step,
· panel size,
., .
· grin spacing,
· vorticity reduction scheme
623

In the present simulations, the flow field is largely
determined by the motion of the cylinder. For this
reason, some precautions must be made with respect
to the coupling of the flow field calculation and the
motion of the cylinder. In order to prevent the sys-
tem from decoupling, the time integration and the
start-up time must be chosen correctly. In simula-
tions without a start-up time, the system became un-
stable in the lock-in region for a dimensionless time
step at* = U^t/D larger than 0.05, i.e. the cylinder
disappeared out of the computation domain which
was 8 cylinder diameters in both directions. A grid
of 64x64 was used in all calculations, with 64 panels
on the cylinder surface. A start-up time of I* = 20
was chosen, see e.g. Sarpkaya t20~. In order to keep
the number of blobs within reasonable limits, i.e. less
than 40000, no experiments could be performed with
a smaller time step, unless very crude vorticity reduc-
tion was applied.
Earlier calculations pointed out that the vorticity
reduction should not be applied in the close vicinity
of the cylinder, to prevent loss of accuracy. The re-
duction or clustering process must be applied with
great care, since it distorts the flow field, and there-
~f - '''
-
Cu rrent
vel oc i ty
L
~ Y
Fig. 2. Vertical riser arrangement
x
Table 3
fore the shedding process. A bi-cubic interpolation
to a fixed grid sufficiently far away from the cylinder
did not cause dramatic disturbances in the motion of
the cylinder, although it was noticable in the force
registration.
Sensitivity to changes in grid and panel size were
not investigated in this study. Numerical results con-
cerning these parameters can be found in e.g. Van
der Vegt A.
The calculations discussed above were performed
on an ETA-1OP232, and a typical simulation required
20 to 30 hours of computing time.
8. Practical Results
In this section some practical results obtained from
calculations with the structural code will be presented.
A first case concerns a vertical marine riser subjected
to a uniform incoming flow. For this application, re-
sults obtained from another computer program are
also available, see Hansen et al. A. The main pa-
rameters are shown in Table 2, while Fig. 2 shows a
general arrangement. Some statistical results of the
vibration properties of the riser are shown in Table 3.
Table 2
Vertical riser properties
Quantity Unit Value
Length m 100
Diameter m 2
Mass kg/m 3220
Natural frequency rad/sec 0.71
Flow velocity m/& 1.0
Tension N 1.6 x 106
Response parameter _ _ _ 0.15
Comparison between calculated results
for vertical riser
Quantity Unit Calculated Hansen .
Xmin m 1.05 1.35
Xma:c m 2.60 2.80
Ymin m 3.05 3.40
Oman m 3.05 3.40
CD _ 1.75 1.60
/\Tma~ N 8.5 x 104 8.0 x 104
624

Typi ca 1 second mode
di spl acement
Extreme di sD1 acements
-2 0 2
In-1 ine displacement (m)
Fig. 3. In-line displacements of vertical riser
Relatively large displacements are shown which
are not unusual for lightly damped structures. The
resemblance with results obtained from Hansen et al.
[1] is good, although their maximum cross-flow and
in-line displacements are larger. This may be caused
by the fact that they use no structural damping at all.
Using the present structural code this is not possible
due to numerical instabilities for very lightly damped
structures.
Figs. 3 and 4 show the extreme in-line and cross-
flow displacements. The in-line displacement is dom-
inated by a static part due to the mean drag force.
The dynamic in-line displacement is dominated by a
second mode vibration. This is due to the frequency
of the drag force which is twice the cross-flow force
frequency. The cross-flow frequency is pronounced
and vibrates in the first excitation mode at a fre-
quency close to the natural frequency.
The second case concerns a horizontal pipe and
cable in a uniform tidal current. Full scale mea-
surement data are provided by Vandiver t244. Fig.
5 shows the experimental arrangement and Table 4
shows the pipe and cable particulars.
The agreement between calculated and measured
vibration data is good, as shown in Table 5.
tic
2n
r
Typical non-symmetrica
di spl acement
Extreme di spl acement
~ or,
l
l
_ I
t
1
,
\
_4 0 4
Cros s-fl ow d i spl acement ( m)
Fig. 4. Cross-flow displacements of vertical riser
Current vel oci ty
in x-direction
x
y
Fig. 5. Horizontal pipe and cable arrangement
Table 4
Horizontal pipe/cable properties
Quantity Unit
Length m
Diameter m
Mass kg/m
Natural frequency rad/sec
Flow velocity m/&
Tension N
Response parameter
| Reynolds number
62s
_ .
Value
pipe
23
0.0414
3.327
4.53
0.5
3500
0.24
18800
cable
23
0.0318
1.952
5.78
0.5
3500
0.27
14500

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